Factor each polynomial completely. See Example 6. 6a² - 11a + 4

Verified step by step guidance

1. Identify the quadratic polynomial in the form \(ax^2 + bx + c\), where \(a = 6\), \(b = -11\), and \(c = 4\).
2. Find two numbers that multiply to \(a \times c = 6 \times 4 = 24\) and add up to \(b = -11\).
3. Rewrite the middle term \(-11a\) as the sum of two terms using the two numbers found in the previous step.
4. Group the terms into two pairs and factor out the greatest common factor (GCF) from each pair.
5. Factor out the common binomial factor from the two groups to write the polynomial as a product of two binomials.

Key Concepts

Factoring Quadratic Polynomials

Factoring quadratic polynomials involves expressing a quadratic expression in the form ax² + bx + c as a product of two binomials. This process helps simplify expressions and solve equations by finding roots or zeros of the polynomial.

Product-Sum Method

The product-sum method is a technique used to factor quadratics where you find two numbers that multiply to ac (the product of the coefficient of a² and the constant term) and add to b (the coefficient of the linear term). These numbers help split the middle term for factoring by grouping.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in a polynomial to factor out common factors from each group. This method is often used after splitting the middle term in a quadratic to factor the expression completely into binomials.